What is the integral from 0 to 2pi of (sin^2(x)) ?

The integral from 0 to 2pi of (sin^2(x)) is pi

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integral from 0 to 2pi of (sin^2(x))

\int_{0}^{2 π} (sin^{2} (x)) d x

π

Decimal Notation

3.14159 \dots

Solution steps

\int_{0}^{2 π} sin^{2} (x) d x

Rewrite using trig identities

= \int_{0}^{2 π} \frac{1 - cos ( 2 x )}{2} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int_{0}^{2 π} 1 - cos (2 x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int_{0}^{2 π} 1 d x - \int_{0}^{2 π} cos (2 x) d x)

\int_{0}^{2 π} 1 d x = 2 π

\int_{0}^{2 π} cos (2 x) d x = 0

= \frac{1}{2} (2 π - 0)

Simplify

\frac{1}{2} (2 π - 0) : π

= π

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\int_{- 2}^{x} (14 t^{2} - 13 t) d t

\int_{- 1}^{1} \frac{3 x ^{2} - 5 x + 1}{4} d x

\int_{0}^{2} 5 x (x^{2} + 1)^{2} d x

\int_{- \frac{π}{2}}^{π} sin (x) d x

What is the integral from 0 to 2pi of (sin^2(x)) ?
The integral from 0 to 2pi of (sin^2(x)) is pi